1. PLANTEAMIENTO DEL PROBLEMA
1.3. JUSTIFICACIÓN
Figure 3.3 shows a typical model of respiratory dead space distribution in human lungs
[26], comprising the common dead space (VDC), regional dead space (VDR) and alveolar gas volume (VA). Following the same approach, the dead space volume in the ventilation system can be divided into two main components, as shown in Figure 3.4(a), illustrating a
schematic layout of the ventilation system used to control the respiratory pattern of the
lung and to deliver an arbitrary mixture of three different types of gases (3He, O2 and air) at any desired ratio, volume and rate. The first dead space component, namely dynamic
dead volume, VD, contains the major conductive airways (trachea and main bronchi) and the portion of the ventilator system after the respirator valve, including endotracheal tube.
Figure 3.3. Partitioning of the lung in common dead space (VDC), regional dead space (VDR) and alveolar
gas volume (VA). The study volume comprises VA and part of the regional dead space (VDS) [26].
Respiratory gas travels towards the lung during inspiration through VD, and away from that during expiration. The second compartment contains part of the ventilator system
that only carries the source gas (i.e. 3He) towards the respirator valve’s inlet. This volume primarily contains the transmission line between the HP gas chamber and the respirator
valve. Unidirectional transport of gas from the source through the transmission line will
3He N2 O2 AIR 3He PS Respirator Exhale ET Tube PIP
Static Dead Volume, VS (Transmission Line)
Dynamic Dead Volume, VD (Conductive Airways) Alveolar Volume (Acinar Airways) MS MT(0) = 0 MT(j) MC(0) = 0 MC(j) MA(0) = 0 MA(j) Tranmission Line rS = VT / VS Conductive Airways rD = VD / VT Acinar Airways r = rA (1-rD) VS VT MS M( j ) M( j – 1) (a) (b) (c)
Figure 3.4. (a) Schematic diagram of the MR-compatible ventilator system depicting the static and dynamic dead space volumes; (b) three-compartment lumped model of ventilator system dead space volumes; and (c) the details of the gas replacement model in the static dead space compartment.
The primary difference between the dynamic and static dead volumes is that the portion
of the respiratory gas residing in VD from the previous breath is re-inhaled with each new breath. Therefore only a fraction of the freshly delivered gas to the lung actually arrives
in the alveolar airways – the remaining fraction being made up of the exhaled gas in the
however, only travels towards the lung and this mixture becomes identical to the source
gas after a finite number of breaths, and therefore is not subject to rebreathing.
A lumped three-compartment model is proposed as illustrated in Figure 3.4(b) to model
the system dead volumes. The rightmost compartment comprises the acinar airways,
including alveoli and small airways, containing the magnetization MA. The middle compartment contains the major conductive airways and both the endotracheal tube and
the respiratory valve that engage in the tidal respiration, containing the magnetization MC. Finally the leftmost compartment includes the transmission line that carries the HP gas
from the source (MS) towards the inlet of the respiratory valve. The magnetization of the gas in the transmission line is referred to as MT, and represents the magnetization of HP gas that enters the respirator valve with each breath.
For the serial ventilation sequence the magnetization buildup in the acinar airways is in
principal governed by Equation [3.6], except that the source magnetization, MS, is now replaced by the magnetization that arrives from the conductive airways at each breath:
€
MA( j) = rA⋅MC( j) + (1−rA)⋅MA( j−1)⋅exp D
[
RF+ DO2]
, MA(0)=0. [3.8]At end-exhale the gas leaving the alveoli and small airways fills the conductive airways.
Therefore the arriving magnetization from the conductive airways, MC, at each breath is a combination of the arriving HP 3He from the transmission line and the exhaled gas from the previous breath. The rebreathing fraction is a function of the dynamic dead space to
tidal volume ratio,
€
rD=VD VT. To ventilate the acinar airways the condition
€
rD<1 has to be met, leading to:
€
MC( j) = (1−rD)⋅MT( j−1) + rD⋅MA( j−1)⋅exp DRF+ DO
2
[
]
, MC(0)=0. [3.9] Substituting Equation [3.9] in Equation [3.8], the relation between the magnetizationbuildup in the acinar airways and the arriving magnetization from the conductive airways
can be expressed as:
€ MA( j) = r⋅MT( j−1) + (1−r)⋅MA( j−1)⋅exp DRF+ DO 2
[
]
, [3.10] where €r = rA⋅(1−rD) is the apparent fractional ventilation, including the rebreathing effect of conductive airways.
At each breath, a tidal volume VT of the source gas (with a nominal magnetization of MS) is driven into the transmission line and mixes with the residual gas MT, present in the static dead volume, VS. Since the gas in the transportation line travels only in one direction towards the respiratory valve, it is assumed that the entrance of HP gas from the
source pushes the same amount of gas (VT) out of the static dead volume (Figure 3.4(c)). Defining
€
rS=VT VS, the mixing of the arriving and residual gases in the static dead space can be recursively expressed as follows:
€ rS<1: MT( j) = rS⋅MS + (1−rS)⋅MT( j−1), MT(0)=0 rS>1: MT( j) = MS, MT(0)=(1−rS−1)⋅MS [3.11]
Equation [3.11] is based on the assumption that the static dead space initially contains no
HP gas, hence the stated initial conditions. This model depicts the relative significance of
the static dead volume in lung ventilation. For a relatively small VT compared to VS (
€
rS<1, e.g. in rodents and small animals), the concentration of the gas delivered from the transmission line MT incrementally increases with each breath. However for large tidal volumes (rS >1, e.g. in humans and large animals), the entire contents of the static dead
space is purged with the first breath, and the magnetization of the following breaths will
be identical to the source magnetization, MS. It is important to note that no decay mechanisms are assumed to affect the accumulated HP gas in the transmission line since
this volume is neither exposed to RF pulses nor the alveolar oxygen gas.